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Planar Motion Control, Coordination and Dynamic Entrainment in Chaplygin

Beanies

Conference Paper · November 2018

DOI: 10.1115/DSCC2018-9037

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Proceedings of of the ASME 2018 Dynamic Systems and Control ConferenceDSCC2018

September 30 - October 3, 2018, Atlanta, Georgia, USA

DSCC2018-9037

PLANAR MOTION CONTROL, COORDINATON AND DYNAMIC ENTRAINMENT INCHAPLYGIN BEANIES

Scott Kelly, Rodrigo Abrajan-GuerreroDepartment of Mechanical Engineering and Engineering Science

University of North CarolinaCharlotte, North Carolina 28223

Email: scott@kellyfish.net, rabrajan@uncc.edu

Jaskaran Grover, Matthew Travers, Howie ChosetThe Robotics Institute

Carnegie Mellon UniversityPittsburgh, Pennsylvania 15217

Email: jaskarag,mtravers,choset@andrew.cmu.edu

ABSTRACTThe Chaplygin beanie is a single-input robotic vehicle for

which partial planar motion control can be achieved by exploit-ing a simple nonholonomic constraint. A previous paper sug-gested a strategy for such motion control. In the present paper,this strategy is validated experimentally and extended to the con-text of multi-vehicle coordination. It is then shown that whenthe plane on which two such vehicles operate is translationallycompliant, energy transfer between the two can enable a mech-anism whereby one (operating under control) may entrain theother (operating passively), partly coordinating their motion. Asan extension to this result, it is further demonstrated that a pairof passive vehicles operating on a translationally compliant plat-form can eventually attain the same heading when released fromtheir deformed configurations.

INTRODUCTIONControl laws that mimic passive physical phenomena are

employed in diverse mechanical settings. An actuator that ap-plies a force proportional to the deviation in a mechanical ele-ment’s measured position from a desired value, for instance, canbe considered to act as a linear spring. Mechanical analogies inturn motivate a variety of control designs. Successful strategiesfor the coordination of multi-vehicle systems, for instance, havebeen developed whereby individual vehicles respond to move-ment relative to their neighbors as if linked to these neighborsby linear springs and dampers [1]. Real physical coupling canalso complement deliberate control in the context of coordina-

tion. Systems of aerial or aquatic vehicles that seek states ofmotion mimicking bird flocks or fish schools, for instance, maydo so using feedback control but may benefit from the local at-tractiveness of states that represent local energetic minima [2,3].The tendency for physically coupled nonlinear systems to syn-chronize may be so strong as to overcome contrary regulation atthe individual level — a point observed as early as Huygens’ ex-periments with synchronizing pendulum clocks in the 1660s [4].

The system depicted in Fig. 1 was introduced in [5] andnamed the Chaplygin beanie because it combines elements oftwo canonical systems from the mechanics literature: the Chap-lygin (or Caratheodory) sleigh (essentially a modern shoppingcart, supported in the front by casters and in the rear by one ormore wheels that roll without slipping [6]) and Elroy’s beanie(a rigid body coupled through an actuator to a balanced rotorsurmounting its center of mass [7]). It was demonstrated analyt-ically in [5] that this system will translate from rest if the rotor isinduced to spin relative to the cart, and that a simple proportionalcontrol law mimicking the action of a torsional spring on the ro-tor can be used to dictate the cart’s asymptotic heading and lon-gitudinal speed simultaneously. An analogy was also developedbetween this system and a planar fishlike swimmer, each able topropel itself by exploiting a resistance to lateral motion at its rearthrough periodic variations in angular momentum forward of thisresistance. A more direct analogy obtains between the Chaply-gin beanie and the aquatic vehicle studied subsequently in [8],consisting of a rigid planar hydrofoil with an internal balancedrotor. In section 1 of the present paper, a physical realization ofthe system in Fig. 1 is presented and used to validate the control

1 Copyright c© 2018 by ASME

θ

φ

a

B

C

(x, y)

FIGURE 1. The Chaplygin beanie. The position of the cart’s centerof the mass and the orientation of the cart relative to a stationary frameof reference are specified by (x,y) and θ , respectively. The rotational in-ertias of the rotor and cart relative to their point of coupling at the cart’scenter of mass are denoted by B and C, respectively. The mass of thesystem overall is denoted by m in the text. The rotational inertia of therear wheel, which can roll freely but cannot slip laterally, is consideredto be negligible.

approach suggested in [5]. In section 2, a strategy is outlined togeneralize this approach to the problem of coordinating the ori-entation and translational speed of two or more Chaplygin bea-nies. In section 3, numerical results are presented that documenta surprising phenomenon whereby a passive Chaplygin beaniewith a torsional spring in place of an actuator will reorient it-self and follow a self-propelling Chaplygin beanie when the twoare supported by a common platform through which vibrationalenergy can be transmitted. Entrainment of this kind is patentlyanalogous to the synchronization of pendulum clocks sharing vi-brational energy through a common mantel, but is also arguablyanalogous to a form of entrainment that occurs within schools offish that exchange kinetic energy through wake vorticity. A dra-matic illustration of the latter is documented in [9]: the flexiblebody of a dead fish can be induced to swim against a backgroundcurrent toward a bluff object upstream when excited by vorticityshed periodically from the object. Finally, in section 4, numericalresults are presented demonstrating entrainment in a pair of pas-sive Chaplygin beanies atop a translationally compliant platform.The entrainment documented in section 3 and 4 is interesting inits own right as a new example of unexpected behavior engen-dered by nonholonomic constraints — the spin reversal of therattleback is a classic, contrasting example [6] — but may alsosuggest a future strategy for coordinating the motion of systemsof underactuated vehicles like the Chaplygin beanie through de-centralized control. F

1 SINGLE-VEHICLE DYNAMICS AND CONTROL1.1 Analysis

The dynamics of the system shown in Fig. 1, with a torqueinducing the rotor to rotate relative to the cart serving as the soli-tary control input, are those of a control-affine system with drift.

The configuration manifold S1× SE(2) is four dimensional, butthe constraint

−xsinθ + ycosθ = aθ

prohibiting lateral slipping of the rear wheel on the ground de-creases the dimension of the system’s phase space by one. Themachinery of nonholonomic reduction [10] was employed in [5]to isolate the dynamics of the two-dimensional nonholonomicmomentum with scalar components

JLT = mxcosθ +mysinθ ,

JRW =−mxasinθ +myacosθ +(B+C)θ +Bφ

representing, respectively, the system’s forward translational mo-mentum and its angular momentum relative to a vertical axispassing through the center of the rear wheel. The system’s driftdynamics are given overall by

x =JLT

mcosθ −a

(JRW −Bα

ma2 +B+C

)sinθ

y =JLT

msinθ +a

(JRW −Bα

ma2 +B+C

)cosθ

θ =JRW −Bα

ma2 +B+C

JLT = ma(

JRW −Bα

ma2 +B+C

)2

JRW =−a(

JRW −Bα

ma2 +B+C

)JLT

φ = α.

(1)

Normalizing with respect to the rotational inertia of the rotor, wecan consider the rotor’s angular acceleration α to be under directcontrol.

A fundamental obstacle to local controllability is presentin (1). The forward momentum JLT is nondecreasing, and anychange in the cart’s heading is necessarily accompanied by an in-crease in this momentum.1 Nevertheless, if the system is initiallyat rest, then it’s possible to induce the cart to accelerate usinga simple proportional controller so that its heading and transla-tional speed — to wit, θ and JLT — will approach any desiredvalues asymptotically. Without loss of generality, the desiredasymptotic value of θ can be taken to be zero. It’s shown in [5](essentially via LaSalle’s invariance principle) that the feedback

1With appropriate magnitude, a sinusoidal torque on the rotor will induce thesystem to accelerate from rest in an undulatory fashion. A movie of this is visibleat http://tinyurl.com/gm4h2nh.

2 Copyright c© 2018 by ASME

law α = kθ will drive θ to zero for any positive value of the gaink. The closed-loop system will evolve, furthermore, so that theenergy-like quantity

Λ =(JRW −Bα)2

ma2 +B+C+

J2LTm

+ kBθ2

is conserved over time. It follows that

limt→∞

J2LT = mΛ(0) = D+ kF, (2)

where

D = m(JRW (0)−Bα(0))2

ma2 +B+C= m(ma2 +B+C)θ(0)2

and

F = mB(θ(0))2.

If θ 6= 0 and θ = 0 initially, then k can be selected to assignany positive value to limt→∞ JLT . If θ 6= 0 but θ 6= 0 initially,then k can be selected to assign any positive value to limt→∞ JLTthat’s greater than a certain lower bound.

1.2 Experimental resultsFig. 2 depicts a physical realization of the Chaplygin beanie.

This system’s dynamics deviate from those of the model (1) prin-cipally because of friction associated with the rear wheel, resist-ing both pivoting of the cart (due to friction between the wheeland the ground) and forward rolling (due to friction in the bear-ings between the cart and the wheel). We demonstrate experi-mentally that if the system in Fig. 2 is initially at rest and its rotoris driven thereafter according to a feedback law akin to α = kθ ,then the heading angle θ will approach zero over time and thecart’s forward speed will approach a value that increases with in-creasing k. The controller we implement is merely “akin to” theproportional controller from section 1.1 in two practical ways.First, the actuator coupling the rotor to the cart is a small brushedDC gearmotor2 powered by a pulse-width modulated (PWM) in-put voltage. The feedback gain k multiplies the cart’s heading todetermine not the motor torque directly but the duty cycle of thePWM signal. Second, limitations on the speed and torque avail-able from the motor confine our experiments to a regime in whichfriction plays a visible role. One way to overcome the cart’s ten-dency to lose forward momentum while stabilizing its heading

2https://www.pololu.com/product/2386

FIGURE 2. A physical Chaplygin beanie constructed primarily fromone inch–thick extruded polystyrene foam. In place of casters, the alu-minum pie plates hover on cushions of air generated by counter-rotatingducted fans to support the forward end of the cart with minimal fric-tion between the cart and the ground. Batteries contribute significantlyto the inertia of the cart; the rotor’s large diameter helps to increase itsrotational inertia without increasing the system’s mass overall.

is to increase its tendency to overshoot this heading, since oscil-lations in heading generate a propulsive force according to thethird and fourth lines in (1). We therefore augment the propor-tional feedback of section 1.1 with an integral feedback term toincrease overshoot, scaling the integral gain with the proportionalgain from one experiment to the next. It’s expected that futureversions of the physical apparatus in Fig. 2 will permit exper-iments involving more aggressive maneuvers that downplay theshort-term influence of friction, obviating the need for integralfeedback. On the other hand, it’s expected that additional ex-periments with the current apparatus will contribute new under-standing to the phenomenon of damping-induced heading recov-ery, a manifestation of damping-induced self-recovery [11] doc-

3 Copyright c© 2018 by ASME

umented in the context of dissipative aquatic locomotion in [12].Figure 3 depicts the results of three different experiments

with the system from Fig. 2. In each experiment, feedback link-ing the angular acceleration of the rotor to the cart’s heading in-duces the cart to execute a right-angle turn. The translationalspeed attained by the cart in doing so increases with increasinggain.

FIGURE 3. Trajectories of the physical Chaplygin beanie under theinfluence of three different heading controllers. In each case, the cartis initially centered at rest along the x axis and pointing in the positivex direction. A PI controller is activated to stabilize the heading to thepositive y direction, resulting in forward locomotion as well as rotation.The dots along the trajectories represent equal intervals of time, show-ing that an increase in feedback gain results in an increase in the cart’stranslational speed. The desired average heading is eventually achievedin every case.

2 HEADING AND SPEED COORDINATIONImagine two identical Chaplygin beanies at rest, situated so

that their initial headings differ. If proportional heading controlis applied to each to steer it toward the initial average heading ofthe two, and if the same gain k is used in both feedback loops,

then the two will approach a state asymptotically in which theytranslate in the same direction at the same speed. The quantity

Λdecoupled =(JRW 1−Bα1)

2 +(JRW 2−Bα2)2

ma2 +B+C

+JLT

21 + JLT

22

m+ kB

(θ

21 +θ

22)

will be conserved as their dynamics evolve, furthermore, even iftheir initial average heading doesn’t correspond to θ = 0.

An alternative strategy for coordinating the asymptotic head-ing and speed of the two vehicles is to implement the pair ofcontrol laws

α1 = k(θ1−θ2), α2 = k(θ2−θ1) (3)

so that each vehicle attempts to stabilize its heading to that ofthe other as the two headings evolve. A comparison of the twostrategies appears in Fig. 4. Using the same shared gain k in bothcases, the second strategy results in more rapid convergence tothe system’s final state. In the second case, the quantity

Λcoupled =(JRW 1−Bα1)

2 +(JRW 2−Bα2)2

ma2 +B+C

+JLT

21 + JLT

22

m+ kB(θ1−θ2)

2

is conserved. The quantities Λdecoupled and Λcoupled differ in the“potential energy” terms that reflect the sense in which the con-trol torques on the two rotors may be associated metaphoricallywith torsional springs.

Recall from (2) that the final translational speed of a soli-tary Chaplygin beanie subject to proportional heading control de-pends on both the feedback gain and the initial error in its head-ing. Because of this, any number of such vehicles with distinctinitial headings can be induced to approach any common headingand translational speed asymptotically through separate control.Though the details aren’t presented here, the authors propose thata set of feedback laws interconnecting the dynamics of a collec-tion of such vehicles in a manner that generalizes (3) can alwaysbe constructed to achieve the same objective, potentially moreefficiently, based on the shaping of an potential energy–like terminvolving a distribution of gains. This will be the topic of a futurepaper.

3 VIBRATIONAL ENTRAINMENTAgain imagine two identical Chaplygin beanies at rest, each

situated and oriented arbitrarily. Suppose that the rotor atop one

4 Copyright c© 2018 by ASME

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FIGURE 4. Trajectories of two Chaplygin beanies subject to pro-portional control stabilizing their headings to a common value. Thetop panel depicts the pair’s response to the decoupled control lawsαi = k(θi − θdesired), the bottom panel to the control laws (3). Bothpanels correspond to the interval 0 < t < 20 and m = a = B =C = k = 1for both vehicles.

cart is actuated but the rotor atop the other is coupled to thecart beneath through a linear torsional spring. Suppose, further-more, that the two vehicles rest atop a common platform thatisn’t rigidly fixed but instead exhibits translational compliancewith finite inertia. It’s clear from the analysis of section 1.1 thatif the rotor atop the first cart is induced to pivot sinusoidally, thenthe cart will advance relative to the platform, exerting a time-varying force on the platform that will induce the platform tomove as well. It’s also relatively clear that if the platform beginsto vibrate, this will induce the second cart to pivot about its rearwheel, in turn exciting the second rotor to oscillate, stimulatingforward propulsion.

What’s unapparent is whether or not a relationship exists be-tween the actuated vehicle’s eventual direction of translation andthat of the passive vehicle. At this point in the narrative, the sys-tem resembles an asymmetric variant of Huygens’ clocks: twooscillators share a substrate through which vibrational energy canbe transmitted, and as a result the oscillatory dynamics of onedrive similar oscillatory dynamics in the other. The reader mightanticipate that the passive vehicle will be attracted to a state of

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FIGURE 5. Initial trajectories leading to the entrainment of a spring-driven Chaplygin beanie (orange) by a sinusoidally actuated Chaplyginbeanie (blue) on a shared platform that exhibits finite translational iner-tia. The trajectories shown are measured relative to the platform. Theparameters m, a, B, and C are set to unity for both vehicles, as are themass of the platform and the stiffness of the torsional spring couplingthe orange rotor and cart. The blue rotor pivots relative to the blue cartso that φ = sin t. The two vehicles are shown in their initial configu-rations in each panel. Since the forward translational speed of each isnecessarily nondecreasing, a trajectory that appears to indicate trans-lation in reverse actually reflects reorientation followed by translation.Each panel represents the interval 0 < t < 20.

translation parallel or antiparallel to the translation of the actu-ated vehicle, observing that the rear wheel of the passive vehiclewill be maximally responsive to lateral forcing and minimally re-sponsive to longitudinal forcing, but this reasoning suggests nobias between parallel and antiparallel states.

In fact, the passive vehicle will eventually follow the actu-ated vehicle even if doing so requires a complete reversal in theformer’s heading. The asymptotic difference between the aver-age heading of the one vehicle and the average heading of theother isn’t zero — we compare average headings because bothvehicles describe persistently undulatory paths — but remainssmall regardless of the initial position and orientation of the onerelative to the other. This is illustrated in Figs. 5 and 6. Fig.5 depicts initial trajectories corresponding to four distinct initialconditions. Some of these trajectories suggest a rapid conver-gence of the two vehicles’ headings, others do not. Fig. 6 de-picts the long-term average difference in the vehicles’ headingsas a function of the initial difference for two different initial rel-ative positions. This long-term average difference is less thanπ/20 radians in almost every case, indicating a robust tendency

5 Copyright c© 2018 by ASME

FIGURE 6. Asymptotic average relative orientation as a function ofinitial relative orientation for the pair of Chaplygin beanies (one activeand one passive) as depicted in Fig. 5

.

for the actuated vehicle to entrain the passive vehicle in a man-ner suggestive of the entrainment that can occur when one bodypropels itself past another in a fluid.

4 ENTRAINMENT IN PASSIVE BEANIESAs a further extension to the scenarios presented before,

imagine now a situation with two passive beanies resting atopa translationally compliant platform, except that now both bea-nies are coupled to their respective rotors with linear torsionalsprings. In this system, there is no active controller commandingthe movement of the rotor in either beanie. We would like to ex-plore whether it is possible to induce synchronized motion in thetwo beanies by releasing them on the platform from a non-zeroinitial deformation in the torsional springs. As one can imagine,the potential energy stored in the deformed springs would leakinto kinetic energy as the springs unwind, which would result inforward translation of individual beanies. We would like to ex-plore the conditions under which the coupling between the twobeanies is sufficiently strong so as to induce synchronized motionbetween them.

Much like the case where two beanies were both com-manded by proportional heading controllers attempting to alignthe heading of each beanie towards that of the other’s (Eq.(3)),one would expect here that depending on the initial difference inthe headings and a parameter describing the coupling, both bea-nies will translate in completely parallel or antiparallel directionsin the steady state. In the case of active control of two beanies,the coupling parameter was governed by the gain of the propor-tional control law. In this case, the relative magnitude of themass of the platform and mass of a beanie govern the asymptotic

FIGURE 7. Asymptotic average relative orientation as a function ofinitial relative orientation for the pair of passive Chaplygin beanies atopa translationally compliant platform. For this simulation, the mass ofthe platform relative to mass of the beanie is Mplat f orm/mbeanie = 0.01

.

difference between the average headings of the individual bea-nies. In Figure 7, we show the asymptotic difference betweenthe average headings of the two passive beanies as a functionof the initial difference. For this simulation, one of the beanie’sinitial heading was held fixed while the other beanie’s orienta-tion was swept through a 2π range. Once again, the asymp-totic difference in the average heading is bounded above by π

20i.e. lim

t−→∞|θ1(t)−θ2(t)| ≤ π

20 . We are currently investigating ananalytical formalism using a Lyapunov function to analyticallyprove that such a bound exists for this autonomous system.

Additionally, the mass of the supporting platform plays arole in the exchange of energies between the beanies. Greater in-ertia of the platform for a fixed beanie mass and rotor inertia willlimit the synchronization. Indeed as can be seen in Figure 8, asthe mass of the platform relative to the mass of the beanies is in-creased beyond a critical value of 0.155, the heading entrainmentbreaks, reminiscent of a bifurcation.

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and Orientation Control Using Projected Collinear TensegrityStructures,” in Proceedings of the American Control Confer-ence, 2009, pp. 610–615.

[2] S. J. Portugal, T. Y. Hubel, J. Fritz, S. Heese, D. Trobe,B. Voelkl, S. Hailes, A. M. Wilson, and J. R. Usherwood, “Up-wash exploitation and downwash avoidance by flap phasing inibis formation flight,” Nature, vol. 505, pp. 399–402, 2014.

[3] S. Marras, S. S. K. amd J. Lindestrom, D. J. McKenzie, J. F.Steffensen, and P. Domenici, “Fish swimming in schools save

6 Copyright c© 2018 by ASME

FIGURE 8. Effect of increasing the mass of the platform relative tothe mass of individual beanies on the asymptotic headings. For thissimulation, θ1(0) = 0,θ2(0) = π

5.

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[9] D. N. Beal, F. S. Hover, M. S. Triantafyllou, J. C. Liao, andG. V. Lauder, “Passive propulsion in vortex wakes,” Journalof Fluid Mechanics, vol. 549, pp. 385–402, 2006.

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[12] S. D. Kelly and R. Abrajan-Guerrero, “Propulsive head-ing control and damping-induced heading recovery for a free

hydrofoil with an internal rotor,” in Proceedings of the 2016American Control Conference.

[13] J. P. Ostrowski, “The Mechanics and Control of Undula-tory Locomotion,” Ph.D. dissertation, California Institute ofTechnology, 1995.

[14] J.S Grover, J. Zimmer, T. Dear, M. Travers, S. D. Kellyand H. Choset, “Geometric Motion Planning for a Three-LinkSwimmer in a Three-Dimensional Low Reynolds-NumberRegime,” to appear in Proceedings of the 2018 American Con-trol Conference.

7 Copyright c© 2018 by ASME

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